How to move multiple objects into another collection?
You just have to .unlink(object) each object from all linked collections and then .link(object) each object to your target collection. 

Example on how to move all objects in selection to MyCollection:



import bpy

C = bpy.context

# List of object references
objs = C.selected_objects

# Set target collection to a known collection 
coll_target = C.scene.collection.children.get("MyCollection")

# Set target collection based on the collection in context (selected) 
#coll_target = C.collection

# If target found and object list not empty
if coll_target and objs:

    # Loop through all objects
    for ob in objs:
        # Loop through all collections the obj is linked to
        for coll in ob.users_collection:
            # Unlink the object
            coll.objects.unlink(ob)

        # Link each object to the target collection
        coll_target.objects.link(ob)